Santa Clara University
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Title: p-adic companion forms for Yoshida lifts
Abstract: Coleman showed that the (k-1)st power of the theta operator q d/dq defines a map from overconvergent modular forms of weight 2-k and slope 0 to weight k and slope k-1. Moreover, the critical p-stabilization of a classical CM form is the image of a p-adic CM form, strengthening the fact that its Galois representation splits locally at p. In the GSp4 setting, the Galois representation of a Yoshida lift splits locally into two 2-by-2 blocks at p. We prove an analogous strengthening in the joint work in progress with Bharathwaj Palvannan. The relevant theta operator arises from the last differential of the dual BGG complex. We computed its explicit effect on q-expansions. Using the explicit Fourier coefficients of Yoshida lifts by Hsieh--Namikawa, we show that Yoshida lifts lie in the image of this theta operator, with a certain choice of p-stabilization.